Piecewise functions worksheet with three problems to graph and evaluate.

Let’s solve each problem step by step.

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Problem 1:

Function:
f(x) = { 2x + 3, if x < 0
{ 3 - x, if x ≥ 0

We need to evaluate:
- f(0): Since 0 ≥ 0, use the second part → 3 - 0 = 3
- f(-3): Since -3 < 0, use the first part → 2*(-3) + 3 = -6 + 3 = -3
- f(2): Since 2 ≥ 0, use the second part → 3 - 2 = 1

✔ All correct.

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Problem 2:

Function:
f(x) = { x + 3, if x ≤ 0
{ 3, if 0 < x ≤ 2
{ 2x - 1, if x > 2

Evaluate:
- f(-1): Since -1 ≤ 0 → -1 + 3 = 2
- f(1): Since 0 < 1 ≤ 2 → use middle piece → 3

✔ Correct.

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Problem 3:

Function:
f(x) = { 3x + 5, if x ≤ -2
{ x - 4, if x > 2

Note: There is a gap between x = -2 and x = 2 — no definition for -2 < x ≤ 2. But we’re only asked to evaluate at points that are defined.

Evaluate:
- f(2): Wait! The function says “if x > 2” for the second piece. So x=2 is NOT covered? Let’s check again.

Actually, looking back:

Original problem 3:
f(x) = { 3x + 5, x ≤ -2
{ x - 4, x > 2

So what about x = 2? It’s not in either condition. That means f(2) is undefined? But the worksheet asks to evaluate it — maybe there’s a typo?

Wait — let me double-check the image description you gave. You wrote:

> 3. f(x) = {3x + 5, x ≤ -2
> { x - 4, x > 2

But then it asks to evaluate f(2), f(-3), f(0).

f(-3): -3 ≤ -2 → 3*(-3)+5 = -9+5 = -4

f(0): 0 is not ≤ -2 and not > 2 → so technically undefined? But perhaps the problem meant to include x > -2 or something else? Or maybe it's a trick?

Wait — maybe I misread. Let me re-express:

The function is only defined for x ≤ -2 OR x > 2. So:

- f(2): 2 is not > 2 → undefined? But the worksheet expects an answer. Hmm.

Perhaps there’s a mistake in how I’m reading. Maybe the second piece is supposed to be x ≥ 2? But as written, it’s x > 2.

Alternatively, maybe the student is expected to realize that f(2) is not defined? But that seems unlikely for this level.

Wait — let’s look again at your original text:

You wrote:

> 3. f(x) = {3x + 5, x ≤ -2
> { x - 4, x > 2

And then:

> Evaluate the following:
> f(2) =
> f(-3) =
> f(0) =

This is problematic because neither piece covers x=2 or x=0.

But perhaps in the actual image, the second piece was meant to be x ≥ 2? Or maybe it’s x > -2? Without seeing the image, I have to go by what’s written.

However, since this is a homework worksheet and they’re asking to evaluate these, likely there’s a typo in transcription.

Looking at common patterns, often piecewise functions cover all real numbers. Here, between -2 and 2 (inclusive of 2?), it’s missing.

But wait — maybe the second piece is actually for x > -2? That would make more sense.

Alternatively, perhaps the second piece is x ≥ 2? Then f(2) = 2 - 4 = -2.

Given that the worksheet includes f(2) and f(0), and assuming it’s solvable, I think there might be a transcription error.

But since you provided the text as:

“3. f(x) = {3x + 5, x ≤ -2
{ x - 4, x > 2”

Then strictly speaking:

- f(2): undefined (since 2 is not > 2)
- f(-3): 3*(-3)+5 = -4
- f(0): undefined

But that can’t be right for a standard homework.

Wait — perhaps the second piece is meant to be x ≥ -2? No, that doesn't fit.

Another possibility: maybe the second piece is for x > -2? Then:

If f(x) = { 3x+5, x≤-2
{ x-4, x>-2 }

Then:

f(2) = 2-4 = -2
f(-3) = 3*(-3)+5 = -4
f(0) = 0-4 = -4

That makes sense and covers all values.

I suspect the original problem had a typo in your transcription, and the second piece should be for x > -2, not x > 2.

Because otherwise, f(0) and f(2) are undefined, which is unusual for such worksheets.

Given context, I’ll assume it’s a typo and the second piece is for x > -2.

So revising:

Assume:
f(x) = { 3x + 5, x ≤ -2
{ x - 4, x > -2 }

Then:

- f(2) = 2 - 4 = -2
- f(-3) = 3*(-3) + 5 = -9 + 5 = -4
- f(0) = 0 - 4 = -4

This is consistent and likely what was intended.

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Final Answers:

Problem 1:
f(0) = 3
f(-3) = -3
f(2) = 1

Problem 2:
f(-1) = 2
f(1) = 3

Problem 3 (assuming correction):
f(2) = -2
f(-3) = -4
f(0) = -4

But to match exactly what was written, if we take literally:

Problem 3 as written has gaps, but since the worksheet asks for answers, I'll go with the corrected version.

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Final Answer:

1. f(0) = 3, f(-3) = -3, f(2) = 1
2. f(-1) = 2, f(1) = 3
3. f(2) = -2, f(-3) = -4, f(0) = -4